Blue is a separate gene that works only with the black gene. It is what is called a "modifier", meaning it will alter the color of black. It is also an incomplete dominant, meaning that instead of needing two copies of a gene to affect the black color, only one is needed. Two copies will make the bird an even lighter color blue, called a "splash".

In order for you to get a blue chicken, you have to have one parent that already has the blue gene. You cannot get it by breeding a black to a white, because black and white are separate genes, and you cannot mix them like, say, paint. It just doesn't work that way. Because the blue gene works only with the black gene, the other parent needs to be either black or blue also. The blue chicken has a 50% chance of passing the blue gene on to it's offspring since it only has one copy, so if mated to a black chicken, half of it's offspring will be blue, the other half black. If two blue chickens are mated, 50% of the offspring will be blue, the other 50% splash, since both parents can potentially contribute a blue gene. Splash has two copies of the blue gene, so if two splash chickens are mated, all their offspring will come out splash.

According to Ideal,
When mating 2 blues you offspring will be 2 blues, 1 black, 1 splash. (Keep in mind this doesn't mean out of 4 babies you will always get this same ratio)
However, breeding splash and black will give you 100% blues.
Look at Ideal Poultry at any kind of blue chicken and the genetics will be explained.

You can get blues from black x white, but only if the white is carrying blue. That just means that if a bird is white, say dominant white, but also blue, you cant tell he is blue because the white is hiding it. White pretty much hides everything.

I've put this in a couple of other posts, but white is not really a color-it's more like a drape over a another color, blotting out whatever is underneath. Breeding whites to other colors can tell you what is underneath. When I was younger I have a black silkie hatch out a blue-she had been in with a white cochin bantam roo. Loved that little crossbred pullet-PERFECT Andalusian Blue color and lacing. wish I could get purebreds to look that good.